A pointwise constrained version of the Liapunov convexity theorem for single integrals

Given any AC solution ${overline{x} : [a,b] rightarrow mathbb{R}^{n}}$ to the convex ordinary differential inclusion $$x^{prime} ( t) in co{v^{1} ( t), ldots, v^{m} ( t)} qquad a.e. on [ a,b], qquad qquad (^{*})$$ we aim at solving the associated nonconvex inclusion $$x^{prime} ( t) in {v^{1} ( t), ldots, v^{m} ( t)} qquad a.e.,x( a) = overline{x} ( a), x( b) = overline{x} ( b), qquad qquad (^{**})$$ under an extra pointwise constraint (e.g. on the first coordinate): $$x_{1} ( t) leq overline{x}_{1} ( t) qquad forall t in [ a,b]. qquad qquad qquad (^{***})$$ While the unconstrained inclusion (**) had been solved already in 1940 by Liapunov, its constrained version, with (***), was solved in 1994 by Amar and Cellina in the scalar n = 1 case. In this paper we add an extra geometrical hypothesis which is necessary and sufficient, in the vector n > 1 case, for it existence of solution to the constrained inclusion (**) and (***). We also present many examples and counterexamples to the 2 × 2 case.